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p-adic étale cohomology of period domains
Pierre Colmez, Gabriel Dospinescu, Julien Hauseux, Wieslawa Niziol
To cite this version:
Pierre Colmez, Gabriel Dospinescu, Julien Hauseux, Wieslawa Niziol. p-adic étale cohomology of pe- riod domains. Mathematische Annalen, Springer Verlag, 2021, 381 (1-2), pp.105-180. �10.1007/s00208- 020-02139-6�. �hal-03024255�
PIERRE COLMEZ, GABRIEL DOSPINESCU, JULIEN HAUSEUX, WIESŁAWA NIZIOŁ
Abstract. We compute thep-torsion andp-adic étale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. As in the cases of`-torsion or`-adic coefficients,`6=p, considered by Orlik, the results involve generalized Steinberg representations.
For thep-torsion case, we follow the method used by Orlik in his computations of the`-torsion étale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For thep-adic case, we don’t use Huber’s definition of étale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous étale cohomology with compact support.
Contents
1. Introduction 1
2. Extensions between generalized Steinberg representations 7
3. The Kottwitz set B(G) 22
4. Period domains 26
5. The geometry of complements of period domains 30
6. The main result 37
Appendix A. Adic potpourri 45
References 49
1. Introduction
Let p be a prime number. One of the main results of [12] and [13] is the computation of the geometricp-adic étale cohomology of Drinfeld p-adic symmetric spaces in arbitrary dimension. The final result is analogous to the one in the case of `-adic étale cohomology with ` 6= p, which was known by the work of Schneider and Stuhler [62]. The Drinfeld symmetric spaces are among the most classical examples of p-adic period domains but it is well-known that they are very special1. In fact, the proofs in [12] and [13] use these unique properties of Drinfeld spaces hence it was not clear to us whether the results of loc. cit. would extend to more general p-adic period domains.
The purpose of this paper is to show that, forcompactly supported p-torsion étale cohomology, it is possible to treat fairly generalp-adic period domains. Moreover, the result is similar to the one for
`-torsion, `6=p, cohomology with compact support obtained2 by Dat [16] (for the Drinfeld spaces) and by Orlik [51] (in general). That this is the case is a little surprising since, as we will explain below, the p-adic étale cohomology with compact support (in the sense of Huber [36]) of p-adic
Date: September 3, 2020.
The research of J.H. was partially supported by the projects ANR-11-LABX-0007-01 CEMPI and ANR-16-IDEX- 0004 ULNE. The research of P.C., G.D., and W.N. was partially supported by the projects ANR-14-CE25-0002-01 PERCOLATOR and ANR-19-CE40-0015-02 COLOSS.
1See [57, Sec. 3] for a list of such properties.
2The Euler characteristic of period domains was known before, thanks to Kottwitz and Rapoport, see [17] for a beautiful presentation.
1
period domains is not at all similar to its `-adic counterpart, ` 6= p, computed by Orlik [53], and seems to produce not very useful objects. On the other hand, the continuous compactly supported cohomology that we define gives reasonable objects (at least in the situation we consider or in the case of the complement of a subvariety in a proper analytic variety as considered3in [44]).
While the arguments in [12] and [13] are based onp-adic Hodge theory (via the syntomic method) and its integral versions [4, 5, 9], this paper combines a beautiful geometric construction due to Orlik [51] with a vanishing result for extensions between mod prepresentations of p-adic reductive groups. The proof of the second result is the main difference with the`6=pcase. We are not able to recover the results of [12] and [13] using the methods used here, and conversely the methods in loc.
cit. do not seem to give the results obtained in this paper for Drinfeld spaces (Poincaré duality with p-torsion coefficients holds for “almost proper” analytic varieties [44], but probably does not hold for general analytic varieties, at least in a naive sense). Orlik did recover in [54] the computation of p-adic pro-étale cohomology of Drinfeld spaces from [12] using his method – which is the one of this paper as well – but one encounters considerable technical difficulties4 when working with the étale cohomology instead of the pro-étale one.
1.1. Notation. In order to state the main results of this paper we need to introduce some notation.
Let C be the completion of an algebraic closure ofQp and let (G,[b],{µ}) be a local Shtuka datum over Qp. HereGis a connected reductive group overQp,[b]is an element of the Kottwitz setB(G) of σ-conjugacy classes in5 G( ˘Qp), i.e., an isomorphism classNb of isocrystals withG-structure over Q˘p, and{µ} is a conjugacy class of geometric cocharacters ofG. Moreover, we ask that [b]lies in the Kottwitz set6 B(G, µ), a certain finite subset ofB(G) defined roughly by a comparison between the Hodge polygon attached to µ and the Newton polygon attached to Nb (see [42, ch. 6] for the precise definition of the set B(G, µ)). This assumption is made so that the period spaces whose cohomology want to compute are not empty.
The pair (G,{µ}) gives rise to a generalized flag variety7 F = F(G,{µ}) defined over the field of definition E of {µ}, a finite extension of Qp and a local analogue of the reflex field in the theory of Shimura varieties. We will consider F as an adic space overSpa(E,OE). Letting E˘ =EQ˘p, the p-adic period domainintroduced by Rapoport and Zink [60]
Fwa=Fwa(G,[b],{µ})
is a partially proper open subset of F ⊗E E, classifying the weakly admissible filtrations of type˘ {µ} on the isocrystal Nb. Basic examples of p-adic period domains are the adic affine spaces, the projective spaces, and the Drinfeld symmetric spaces (complements of the union of all Qp-rational hyperplanes in the projective spaces).
As we have already mentioned, Orlik computed in [51] the `-adic compactly supported étale cohomology of these period domains when G is quasi-split over Qp, [b] is a basic class, and ` 6=p is a sufficiently generic prime number. We will also assume that G is quasi-split over Qp and that b ∈ G( ˘Qp) is basic and s-decent8. We refer the reader to the main body of the article for these
3In both cases this continuous compactly supported cohomology coincides with the naive one.
4For example, the rational p-adic pro-étale cohomology of an open ball has a simple description in terms of differential forms [14], but the integrality conditions coming from thep-adic étale cohomology make the computations subtler.
5Q˘pis the completion of the maximal unramified extension ofQpinC.
6For the main result of the paper it would be enough to assume that[b]belongs to the larger setA(G, µ), since all we need is that the period domain is nonempty, which is equivalent to [b]∈A(G, µ)by a result of Fontaine and Rapoport [25].
7IfGis quasi-split overQp, which will be the case in our main result, we can chooseµ∈ {µ}defined overEand thenF=F(G,{µ})is the quotient ofGE by the parabolic subgroupP(µ)associated toµ.
8The hypothesis thatbis decent is harmless, since anyσ-conjugacy class inG( ˘Qp)contains ans-decent element for some positive integers≥1.
notions, introduced by Kottwitz (for the first one) and Rapoport-Zink (for the second one). This implies, for instance, that b ∈ G(Qps) and that the period domain Fwa = Fwa(G,[b],{µ}) has a canonical model (still denoted Fwa) over Es =EQps ⊂Qp. Let Jb be the automorphism group of Nb. It is a connected reductive group over Qp, which is an inner form ofG (this is equivalent to b being basic). The natural action ofG( ˘Qp) on the flag varietyF⊗EE˘ induces an action ofJb(Qp) on the period domain Fwa. In particular, we obtain an action ofJb(Qp)×GEs,GEs = Gal(Qp/Es), on H´et,c∗ (FCwa,Z/`n) and H´et,c∗ (FCwa,Z`), for any prime `. The main theorem of this paper gives a simple description of these representations in the case `=p.
Let T be a maximal torus of G such that µ factors through T, and let W = N(T)/T be the (absolute) Weyl group of G with respect to T, which acts naturally on X∗(T). Let Wµ be the set of Kostant representatives with respect to W/Stab(µ), i.e., the representatives of shortest length in their cosets. The group GEs acts on W and preserves Wµ since µ is defined over Es. One can associate to eachGEs-orbit[w]∈Wµ/GEs the following objects:
• An integerl[w], the length of any element of [w].
• For any prime `, a Z/`n[GEs]-module ρ[w](Z/`n), which is simply the Z/`n-module of Z/`n- valued functions on [w], with the obvious GEs-action twisted (à la Tate) by −l[w].
We will simply write J instead of Jb from now on. Choose a maximal Qp-split torus S of Jder contained in T and a minimalQp-parabolic subgroupP0 of J containingS. Let ∆⊂X∗(S) be the associated set of relative simple roots. For each subset I of ∆, we denote by PI the corresponding standard Qp-parabolic subgroup ofJ, so that P∅ =P0 and P∆ =J. Consider the compact p-adic manifold
XI=J(Qp)/PI(Qp).
If R is an abelian group, let
vPJI(R) =iJPI(R)/X
I(I0
iJPI(R), iJPI(R) = LC(XI, R),
be the corresponding generalized Steinberg representation of J(Qp), with coefficients in R (here LC(?, R) is the space of locally constant functions on? with values inR).
Finally, choose an invariant inner product (−,−)on G, i.e., an inner product onX∗(T0)⊗Q, for all maximal toriT0 ofG, compatible with the adjoint action ofG(Qp)and the natural action ofGQp
on maximal tori ofG. It induces an invariant inner product on J as well. For each GEs-orbit[w]of w∈Wµ, define
I[w]={α ∈∆|(wµ−ν, ωα)≤0}, P[w]:=PI[w], whereωα ∈X∗(S)⊗Q,α∈∆, form the dual basis of∆.
1.2. The main result. Recall that, for `6=p, we have the following computation of Orlik.
Theorem 1.1 (Orlik [51, 53]). Let (G,[b],{µ}) be a local Shtuka datum with G/Qp quasi-split, b∈G( ˘Qp) basic ands-decent. Let `6=p be sufficiently generic9 with respect to G.
There are isomorphisms of GEs ×J(Qp)-modules H´et,c∗ (FCwa,Z/`n)' M
[w]∈Wµ/GEs
vPJ
[w](Z/`n)⊗ρ[w](Z/`n)[−n[w]], H´et,c,Hu∗ (FCwa,Z`)' M
[w]∈Wµ/GEs
vPJ
[w](Z`)⊗ρ[w](Z`)[−n[w]],
where n[w] = 2l[w]+|∆KI[w]| and H´et,c,Hu∗ denotes Huber’s compactly supported cohomology. In particular, the action of J(Qp) on H´et,c,Hu∗ (FCwa,Z`) is smooth.
Our main result is the following computation.
9See [51, Sec. 1] for the definition.
Theorem 1.2. Let (G,[b],{µ})be a local Shtuka datum withG/Qp quasi-split, b∈G( ˘Qp)basic and s-decent. Assume that p≥5.
There are isomorphisms of GEs ×J(Qp)-modules H´et,c∗ (FCwa,Z/pn)' M
[w]∈Wµ/GEs
vPJ[w](Z/pn)⊗ρ[w](Z/pn)[−n[w]], H´et,c∗ (FCwa,Zp)' M
[w]∈Wµ/GEs
vPJ,cont
[w] (Zp)⊗ρ[w](Zp)[−n[w]], where H´et,c∗ denotes the continuous compactly supported cohomology, vJ,contP
I (Zp) = lim
←−nvPJ
I(Z/pn) denotes continuous Steinberg representations, andρ[w](Zp) = lim
←−nρ[w](Z/pn).
Remark 1.3. The result for torsion coefficients in Theorem 1.2 is analogous to the one of Orlik quoted above. The analog of Orlik’s second isomorphism is false: in the case of the adic affine space A1Qp, which is a period domain for the groupG=Gm,Qp×Gm,Qp, we obtain (in the appendix) the isomorphism
H´et,c,Hu2 (A1C,Zp(1))'(OP1
C,∞/C)⊕Zp,
where OP1C,∞ is the stalk of analytic functions at ∞. This result is to be compared with the isomorphism H´et,c,Hu2 (A1C,Z`(1)) ' Z`, for ` 6= p. Note, moreover, that the action of G(Qp) on H´et,c,Hu2 (A1C,Zp(1)) is not smooth.
Remark 1.4. The case of A1Qp suggests that Huber’s definition is not the right one for p-adic coeffi- cients. On the other hand, the continuous compactly supported cohomology10
RΓ´et,c(X,Zp) := R lim
←−n RΓ´et,c(X,Z/pn),
gives sensible results, as Theorem 1.2 shows. In this particular case, we have an isomorphism H´et,ci (FCwa,Zp)'H´et,c,naivei (FCwa,Zp) := lim←−n H´et,ci (FCwa,Z/pn), i≥0,
with the naive version of compactly supported cohomology. We note that in a recent preprint [44], Lan-Liu-Zhu prove a rational Poincaré duality for p-adic étale cohomology of "almost proper" adic spaces and the compactly supported cohomology that they use is the naive one, which is equal to the continuous one in their setting because their torsion cohomology groups are finite (hence satisfy the Mittag-Leffler condition).
Note that one could use also the continuous compactly supported cohomology in the `-adic case,
`6=p, instead of Huber’s version. One would get continuous generalized Steinberg representations instead of smooth ones in Theorem 1.1, which would fit better with the objects appearing in the p-adic Langlands program such as Emerton’s completed cohomology. That would also make a (topological) Poincaré duality possible for the spaces that we consider.
Remark 1.5. Moreover:
(1) We expect that the hypothesisp≥5in Theorem 1.2 is not needed. This hypothesis is made so that we can use Theorem 1.8 below, which almost surely holds for anyp.
(2) Let p ≥ 5. Let HdQp be the Drinfeld symmetric space of dimension d over Qp. Recall that HdQp =PdQpK∪H∈HH, where H is the set ofQp-rational hyperplanes. SetG:=GLd+1,Qp. Theorem 1.2 yields an isomorphism of GQp×G(Qp)-modules
H´et,ci (HdC,Z/pn)'Sp2d−i(Z/pn)(d−i),
10Instead of requiring a proper support for a compatible sequence of global sections we just take sequences of properly supported global sections.
where the generalized Steinberg representations Spj(Z/pn) are as defined in Section 6.1.3.
Comparing this isomorphism with that of [13]:
H´eti (HdC,Z/pn)'Spi(Z/pn)∗(−i)
one finds an abstract duality ofGLd+1(Qp)×GQp-representations:
H´et,ci (HdC,Z/pn)(d)'H´et2d−i(HdC,Z/pn)∗.
It seems likely that this abstract duality is induced by the cup-product with values in H´et,c2d (HdC,Z/pn(d))'Z/pn but we did not verify this.
This suggests that Poincaré duality holds forFCwa and that one can deduce from Theorem 1.2 a description of the étale cohomologyH´et∗(FCwa,Z/pn) asJ(Qp)×GQp-modules.
(3) Suppose moreover thatµis minuscule. Thanks to the work of Fargues-Fontaine [24], Kedlaya- Liu [37], and Scholze [63], we can define theadmissible locus Fa⊂Fwa, a partially proper open subset ofF having the same classical points asFwa, and ap-adic local system over it interpolating the Galois representations associated to these classical points by the theorem of Colmez-Fontaine. In some remarkable situations (which can be completely classified thanks to the work of Chen-Fargues-Shen [10] and Goertz-He-Nie [26]) we have Fwa = Fa and so the above theorem describes the p-adic étale cohomology with compact support of the admissible locus. For instance, this is the case for the quasi-split groupG= SO(V, q), where V =Qnp endowed with the quadratic form q(x1, . . . , xn) =x1xn+x2xn−1+· · ·+xnx1, the minuscule cocharacter µ(z) = diag(z,1, . . . ,1, z−1), and the basic class [b] = [1]∈B(G, µ), for whichJ =G. The flag variety is then the quadric F over Q˘p with equationq(x) = 0 in projective space and we have Fwa =Fa = F KG(Qp)S, where S is the Schubert variety with equations xdn/2e+1 = · · · = xn = 0 inside F (we learnt this example from Fargues).
Forn= 21, we obtain a very concrete description of the p-adic period domain for polarized K3 surfaces with supersingular reduction and the previous theorem yields its p-adic étale cohomology with compact support. In general, we do not know how to describe the `-adic étale cohomology (with compact support) ofFa, even for `6=p.
1.3. The proof of the main result. We will sketch the proof of Theorem 1.2 in the torsion case;
the continuous case follows by taking limits.
1.3.1. The geometric part. As we have already mentioned, the geometric part of the proof is analo- gous to Orlik’s proof of the corresponding result with `-torsion coefficients, for` 6=p. Our contri- bution here lies solely in the verification that all `-torsion statements in Orlik’s proof work in the p-torsion setting as well. That this was not guaranteed is shown by the fact that it fails in the`-adic setting: Orlik’s`-adic proof for`6=pbreaks downp-adically (as we have already seen in remark 1.3, Huber’s compactly supportedl-adic cohomology behaves rather badly forl=p, while it behaves as expected for l6=p, and this is crucial for Orlik’s argument to work).
The argument goes as follows. One starts with the distinguished triangle (associated to the triple (Fwa,F, ∂Fwa),∂Fwa :=F KFwa)
RΓ´et,c(FCwa,Z/pn)−→RΓ´et(FC,Z/pn)−→RΓ´et(∂FCwa,Z/pn).
This reduces the computation ofH´et,c∗ (FCwa,Z/pn) to that ofH´et∗(∂FCwa,Z/pn): one needs to prove an isomorphism (we omit the coefficients Z/pnin the formula):
H´et∗(∂FCwa)'
( L
|∆KI[w]|=1 iJP
[w]⊗ρ[w][−2l[w]] L
L
|∆KI[w]|>1 ρ[w][−2l[w]]⊕ vPJ
[w]⊗ρ[w][−2l[w]− |∆KI[w]|+ 1]
(1.6)
To do it, one stratifies the complement ∂Fwa by Schubert varieties whose cohomology is easy to compute. More precisely, one uses the Faltings and Totaro description of weak admissibility
as a semistability condition: the period domain Fwa is the locus of semistability in F and the complement ∂Fwa is the locus in F, where semistability fails. To test semistability one applies the Hilbert-Mumford criterion: for a field extensionK/Eˇ,x∈F(K)is semistable (hencex∈Fwa(K)) if and only if µ(x, λ) ≥ 0, for all λ∈ X∗(J)GF. Here µ(−,−) is the slope function associated to a linearization of the action ofJ.
The slope function, a priori convex on each chamber of the spherical buildingB(Jder), is actually affine. This implies that, in the Hilbert-Mumford criterion, it is enough to test the 1-parameter subgroups associated to the relative simple roots and their conjugates. This leads to the stratification
∂Fwa=Z1⊃ · · · ⊃Zi−1⊃Zi⊃Zi+1 ⊃ · · ·
that is defined in the following way. For λ∈ X∗(J)Q,let Yλ be the locus inF, where λdamages the semistability condition. ForI ⊂∆, letYI:=∩α /∈IYωα be the associated Schubert variety. Then the locus Zi of ∂Fwa, where the semistability fails to the degree at least i, can be described as
Zi = [
|∆KI|=i
ZI, ZI :=J(Qp)·YIad.
We note that ZI is a closed pseudo-adic subspace of ∂Fwa. In particular, so is ∂Fwa = Z1 = S
|∆KI|=1ZI.
Having this stratification, by a procedure akin to a closed Mayer-Vietoris, one obtains an acyclic complex of sheaves on ∂FCwa, calledthe fundamental complex,
0→Z/pn→ M
|∆KI|=1
(Z/pn)I → M
|∆KI|=2
(Z/pn)I → · · · → M
|∆KI|=|∆|−1
(Z/pn)I→(Z/pn)∅ →0, where(Z/pn)I denotes the constant sheafZ/pnevaluated11onZI,C. This complex yields a spectral sequence
(1.7) E1i,j = M
|∆KI|=i+1
H´eti (∂FCwa,(Z/pn)I)⇒H´eti+j(∂FCwa,Z/pn).
Using the fact that PI(Qp) is the stabilizer ofYI inJ(Qp) and XI =J(Qp)/PI(Qp) one computes that
H´eti (∂FCwa,(Z/pn)I)'LC(XI, H´eti (YI,C,Z/pn))'iJPI(Z/pn)⊗H´eti (YI,C,Z/pn) 'iJPI(Z/pn)⊗( M
[w]∈ΩI
ρ[w](Z/pn)[−2l[w]]).
HereΩI is a subset ofWµ/GEs (see Section 5.5). The third isomorphism is obtained by the classical computation of the cohomology of Schubert varieties. Via a simple Galois-theoretic weight argument, this computation implies that the above spectral sequence degenerates atE2. Using results of Grosse- Klönne [27], Herzig [34], and Ly [46] on generalized Steinberg representations mod p, one can also compute theE2 terms: they are equal to the terms on the right hand side of the formula (1.6).
1.3.2. The group-theoretic part. It follows from the above section that the grading ofH´et∗(∂FCwa,Z/pn) associated to the filtration induced by the spectral sequence (1.7) is isomorphic to the right hand side of (1.6). It remains to show that this filtration splits. And this is where things get much harder for p-torsion coefficients than for the`-torsion ones. Splitting this filtration essentially comes down to understanding Extgroups between generalized Steinberg representations withp-torsion coefficients.
Fortunately, it suffices to deal with Ext1’s, which are the only ones we can handle, contrary to the usual theory with complex coefficients (adapted to the`-adic setting by Orlik [52] and Dat [16]). It is indeed a well-known phenomenon in the theory of smooth modprepresentations ofp-adic reductive
11We simplify for the sake of the introduction; see Section 6.2.1 for details.
groups that Ext groups can be very hard to compute, since most of the techniques for complex or
`-adic coefficients fail.
Before stating the key result that allows us to split the filtration, let us briefly explain the argument for complex or`-torsion coefficients and point out the difficulties occurring forp-torsion coefficients.
Let R be one of the rings C,Z/`n,Z/pn (` being sufficiently generic with respect to J). One can construct an acyclic complex
0→iJ∆(R)→ M
I⊂I0⊂∆
|∆KI0|=1
iJP
I0(R)→ · · · → M
I⊂I0⊂∆
|I0KI|=1
iJP
I0 →iJPI(R)→vJPI(R)→0.
ForR=Cor Z/`n this is a rather standard result, and it also works forZ/pn thanks to the above- mentioned work of Grosse-Klönne, Herzig, and Ly (the acyclicity of this complex is also crucial in computing the E2 terms of the above spectral sequence). Suppose that R 6= Z/pn. A spectral sequence argument reduces the computation of Ext∗J(Q
p)(vPJ
I(R), vPJ
I0(R)) to the computation of Ext∗J(Q
p)(iJP
I(R), iJP
I0(R)) for all I, I0 ⊂∆. The exactness of the Jacquet functor (which fails when R=Z/pn) reduces the problem to understanding extensions between the Jacquet module ofiJP
I(R) (which can be understood by the Bernstein-Zelevinsky geometric lemma) and the trivial represen- tation. After several other relatively standard but technical arguments one reduces everything to the computation of H∗(J(Qp), iJP
I(R)) = H∗(MI(Qp), R), where MI is the Levi quotient of the standard parabolic PI. Thus we are reduced to computing H∗(G(Qp), R) for a reductive group G over Qp, which can be done using the contractibility of the Bruhat-Tits building of Gand the fact thatiGK(1)is injective as smooth representation wheneverK is a compact open subgroup ofG(since passage to K-coinvariants is exact). This again fails when R=Z/pn. Actually, it is an interesting problem to compute H∗(G(Qp),Z/pn) for a reductive group G over Qp. Unfortunately we don’t have much to say about this except to mention that the computation of H∗(GLn(Zp),Z/p) seems rather complicated: Lazard’s theory allows one to compute H∗(1 +pMn(Zp),Z/p) (at least when 1 +pMn(Zp) is a uniform pro-p group, e.g. if p > 2), so one is reduced to the computation of H∗(GLn(Z/p),Z/p), a well-known open problem.
The previous paragraph makes it clear that a new idea is needed in order to compute extensions between generalized Steinberg representations modulo p. We will only focus on the computation of Ext1’s, which is enough for our needs. All Ext groups below are computed in the category of smoothJ(Qp)-representations with coefficients inZ/pn. The key result needed to split the abutment filtration of the spectral sequence (1.7) is then:
Theorem 1.8. Assume that p≥5 and letI, I0⊂∆. If |(I∪I0)K(I∩I0)| ≥2, then Ext1J(Q
p)(vPJI(Z/pn), vPJ
I0(Z/pn)) = 0.
We refer the reader to Section 2.3 for an overview of the rather technical proof of the theorem.
(Actually we compute the Ext1’s between generalized Steinberg representations, with coefficients in an artinian commutative ring in which p is nilpotent, for any reductive group over a local field of residue characteristicp.) The most difficult part is to computeH1(G(Qp),St), whereStdenotes the ordinary Steinberg representation, for a reductive group Gover Qp and one can actually reduce the theorem to this computation by rather painful dévissage arguments involving Emerton’s ordinary parts functor and its derived functors. The computation of higher Ext groups seems much more involved: extending our method would require at least to compute H∗(G(Qp),St) and to prove a conjecture of Emerton (see [20, Conj. 3.7.2]).
Acknowledgments. We would like to thank Sascha Orlik for patiently explaining to us the details of his work. We also thank Laurent Fargues for helpful discussions concerning the content of this paper. G.D. would like to thank Shanwen Wang and the Fudan University, where parts of the paper were written, for the wonderful working conditions.
2. Extensions between generalized Steinberg representations
Let F be a local field of residue characteristic p, G be the group of F-points of a connected reductive algebraic F-group, and R be an artinian commutative ring in which p is nilpotent. We compute theExt1groups between generalized Steinberg representations in the category of smoothG- representations with coefficients inR. In particular, we will prove Theorem 1.8 from the introduction.
2.1. Notation. Let us fix the notation for this section. We fix a separable closureF of F and let GF = Gal(F /F). Let ε : Q∗p → Z∗p denote the p-adic cyclotomic character and let ε¯: Q∗p → F∗p denote its reduction modp.
2.1.1. Linear algebraic F-groups. A linear algebraicF-group will be written with a boldface letter like H and its group of F-points will be denoted by the corresponding ordinary letter H =H(F).
We will write ZH for the center of H.
Let G be a connected reductive algebraic F-group. We fix a maximal split torusS ⊂ Gand a minimal parabolic subgroupB ⊂GcontainingS. LetZZZ be the centralizer of S inG, which is the Levi factor of B containing S, andU be the unipotent radical of B, so that B =ZZZU. We write B =ZZZU for the opposite minimal parabolic subgroup. LetNNN be the normalizer ofS inGand let
W =NNN/ZZZ =N/Z be the relative Weyl group of G. For w∈W, we let
Uw =U ∩w−1Uw and Bw=ZZZUw.
LetX∗(S) be the group of characters ofS, letX∗(S) be the group of cocharacters of S, and let h−,−i:X∗(S)×X∗(S)→Zdenote the natural pairing. Let
Φ⊃Φ+⊃∆
be the subsets of relative roots, positive roots, simple roots inX∗(S). We let Φ− =−Φ+ = ΦKΦ+. For α ∈ ∆, we let α∨ ∈ X∗(S) be the corresponding coroot, sα ∈W be the corresponding simple reflection, and Uα ⊂ U be the corresponding root subgroup. If Uα is one-dimensional, then α extends to a character ofZZZ which will be denoted α.e
For I ⊂ ∆, we let PI = MINI be the corresponding parabolic subgroup of G containing B, where MI is the Levi factor of PI containing S and NI is the unipotent radical of PI, and we let BI =MI∩B (a minimal parabolic subgroup ofMI containingS) andUI=MI∩U (the unipotent radical of BI), so that BI = ZZZUI.12 We write PI = MINI and BI = ZZZUI for the opposite parabolic subgroups. We also let ZI denote the center of MI. We letNNNI be the normalizer of S inMI and we let
WI =NNNI/ZZZ =NI/Z
be the relative Weyl group of MI. We letWfI ⊂W be the set of representatives of minimal length of the right cosetsWI\W, we letwI,0 ∈WI be the longest element, and we let
WcI=wI,0WfIK [
J)I
wJ,0WfJ. ForwI ∈WI, we let
UI,wI =UI∩wI−1UIwI and BI,wI =ZZZUI,wI
When I ={α}(resp. I = ∆K{α}), we rather write Pα,Mα,Nα, etc. (resp.Pα,Mα,Nα, etc.).
12In the caseI=∅we have PI =B,MI =ZZZ,NI =U,BI =ZZZ, andUI ={1}. In the caseI = ∆we have PI=G,MI =G,NI={1},BI=B, andUI=U.
2.1.2. Smooth representations. All representations will be smooth with coefficients inRand all maps between R-modules will beR-linear.
Given a locally profinite spaceX, we letLC(X)be theR-module of locally constant functions on X with coefficients inR, we letsupp(f) =XKf−1({0}) denote the (open and closed) support of a functionf ∈LC(X), and we letLCc(X)⊂LC(X)be theR-submodule consisting of those functions with compact support.
Given a closed subgroup H of G and a smooth H-representation σ, we define a smooth G- representation by letting Gact by right translation on theR-module
IndGH(σ) ={f :G→σ| ∃Kf ⊂G open subgroup s.t.f(hgk) =h·f(g) ∀h∈H, g∈G, k∈Kf}. Let1 be the trivial representation of any locally profinite group. For any subsetI ⊂∆, let
iGPI = IndGPI(1)'LC(PI\G).
If J ⊃ I is another subset, then there is an injection iGP
J ,→ iGP
I which is induced by the natural surjectionPI\GPJ\G. The generalized Steinberg representation with respect toI is the quotient
vPGI =iGPI/X
J)I
iGPJ.
In the case I = ∅we obtain the ordinary Steinberg representation denoted St. In the case I = ∆ we obtain the trivial representation 1.
Given a closed subgroupU0⊂U stable under conjugation byZ, we endow theR-moduleLCc(U0) with a smooth action of the group B0 =ZU0 defined by
(zu¯·f)(¯u0) =f(z−1u¯0z¯u) forz∈Z, u,¯ u¯0 ∈U0, andf ∈LCc(U0).
2.2. The main results. We prove the following result (cf. Theorem 1.8 in the introduction).
Theorem 2.1. Assume thatp≥5and letI, J ⊂∆. If|(I∪J)K(I∩J)|>1, thenExt1G(vPG
I, vGP
J) = 0.
Remark 2.2. We expect Theorem 2.1 to hold true for allp. Actually, we prove the result in almost all cases when p = 3 and in some cases when p = 2. Moreover, it follows from our computations that the above Ext1 is always killed by 3 whenp= 3 and by16 whenp= 2 (see Remarks 2.11 and 2.44). In particular, if char(F) = 0 and E/F is a finite extension, then the analogous Ext1 in the category of admissible unitary continuous G-representations onE-Banach spaces vanishes for allp (see [30, Prop. 5.3.1] and [29, Lemme 3.3.3]).
When I =J, the R-module Ext1G(vGP
I, vPG
J) has been computed in [33] without any assumption (see Proposition 8 in loc. cit.). In the course of the proof of Theorem 2.1, we also treat the case J =It {α} under a very mild assumption (which is always satisfied if p≥5) and, when p6= 2, we reduce the remaining case I =Jt {α} to the special case where∆ ={α}. We treat the latter case under some assumption on G. In particular, we obtain the following result.
Theorem 2.3. Assume that G= GLn(D) for some division algebra Dover F and letI, J ⊂∆. If J 6⊂I assume moreover that D6=Q2.
(1) If J =It {α}, then the R-module Ext1G(vPG
I, vGP
J) is free of rank 1.
(2) If I =Jt {α}, then there is an R-linear isomorphism Ext1G(vPGI, vGPJ)'Hom(E∗, R), whereE denotes the center of D.
(3) If |(I∪J)K(I∩J)| ≥2, then Ext1G(vPG
I, vGP
J) = 0.
In contrast, we do not know how to compute the R-module Ext1G(St,1) when G= GL2(Q2), or Ext1G(vPG
1, vGP
2) when G= GL3(Q2) and P1, P2 denote the two maximal proper standard parabolic subgroups of G.
Remark 2.4. A locally analytic version of part (2) of Theorem 2.3 (whenD=F and char(F) = 0) is established in the work of Ding [18] and generalized to split reductive groups by Gehrmann [45].
Higher Ext groups are computed by Orlik and Strauch [55], for split reductive groups and in a suitable category of locally analytic representations (but not in the category of admissible locally analytic representations). We note that a vanishing result for Ext1 in the locally analytic world gives a corresponding vanishing result in the context of admissible Banach representations, since the continuous generalized Steinberg representations are the universal unitary completions of their locally analytic vectors. We thank Lennart Gehrmann for pointing out the references above. Let us mention though that the vanishing mod p in Theorem 1.8 is crucial for the proof of Theorem 1.2:
the corresponding result for Banach representations is not sufficient for our needs.
2.3. The general strategy. We fix two subsetsI, J ⊂∆. First, we recall the computation of the R-module HomG(vGP
I, vPG
J).
Proposition 2.5 (Grosse-Klönne, Herzig, Ly). There is an R-linear isomorphism HomG(vPGI, vGPJ)'
(R ifI =J, 0 otherwise.
Proof. If I = J, then the result is a special case of [33, Cor. 5]. If I 6= J, then by dévissage the result reduces to the case where R is a field of characteristic p, which is proved by Grosse-Klönne
[27], Herzig [34], and Ly [46].
Now, using the results of Sections 2.4–2.8, we compute theR-moduleExt1G(vGP
I, vGP
J)whenI 6=J.
We treat the two following cases separately.
CaseJ 6⊂I: Let α ∈(∆KI)∩J. If F =Qp and dimUα = 1 assume thatε¯◦αe 6= 1 (this is always true ifp≥5, and actually we only need this assumption when J K{α}=I∩ {α}⊥, see Remark 2.11). There is anR-linear isomorphism (see (2.7)):
Ext1G(vPGI, vGPJ)'
(R if J =It {α}, 0 otherwise.
In the caseJ =It{α}, theR-moduleExt1G(vPG
I, vPG
J)is generated by the class ofIndGPα(vMMαα∩PI) (see (2.6)).
CaseJ (I: Assume that p6= 2 and let us prove the following results.
(1) IfI =Jt {α}and the adjoint action ofZ onUαK{1}is transitive (e.g. ifG= GLn(D) for some division algebra D overF), then there is an R-linear isomorphism
Ext1G(vGP
I, vPG
J)→∼ Ext1Z(1,1)sα=−1
where the right-hand side denotes the R-submodule of Ext1Z(1,1) consisting of those extensions on which sα acts by multiplication by−1.
(2) If |IKJ|>1, thenExt1G(vPG
I, vGP
J) = 0.
We proceed by induction on|∆KI|. If I 6= ∆, then we fix α ∈ ∆KI and we can use the induction hypothesis twice with the exact sequence ofR-modules (see (2.8))
0→Ext1G(vPG
I, vPG
J)→Ext1Mα(vMMαα∩PI, vMMαα∩PJ)→Ext1G(vGP
It{α}, vPG
J).
In the caseI = ∆, we proceed by induction on|J|. IfJ 6=∅, then we fix α∈J and we can use the induction hypothesis twice with the exact sequence ofR-modules (see (2.12))
0→Ext1G(1, vGPJ)→Ext1Mα(1, vPMα
JK{α})→Ext1G(1, vPG
JK{α}).
In the base caseI = ∆and J =∅, the results are given by Propositions 2.45 and 2.47 (this is the most difficult part, which needs the assumptionp6= 2).
This completes the proof of Theorem 2.1. We turn to Theorem 2.3. Assume that G= GLn(D) for some division algebra D over F. In the case J 6⊂ I, the above assumption becomes D 6= Q2. In the case J (I, the above results hold true whenp = 2(see Remarks 2.46 and 2.48 for the base case).
2.4. Reduction to the caseI = ∆ andJ =∅. We recall some results on the parabolic induction functor and its right adjoint. Let I ⊂∆. The parabolic induction functorIndGPI, from the category of smooth MI-representations to the category of smooth G-representations, is exact and preserves admissibility. When char(F) = 0, Emerton [19] constructed a functor OrdP
I (the ordinary part), from the category of smoothG-representations to the category of smoothMI-representations, which is left-exact, preserves admissibility, and is right adjoint of IndGPI when restricted to admissible representations. His construction was generalized by Vignéras [67] to include the case char(F) =p.
Whenchar(F) = 0, Emerton [20] extended the functorOrdP
I (which is not exact as soon asI 6= ∆) to a sequence of functorsHnOrdP
I which preserve admissibility and form aδ-functor when restricted to admissible representations. Whenchar(F) =p, the restriction of the functorOrdP
I to admissible representations is exact (this is due to one of us, see [31, Th. 1]) and we setHnOrdP
I = 0forn≥1.
2.4.1. Reduction to the case I = ∆. Assume that I 6= ∆ and let α ∈ ∆KI. By exactness and transitivity of parabolic induction, there is a short exact sequence ofG-representations
(2.6) 0→vPG
It{α} →IndGPα(vMMαα∩PI)→vGPI →0 which induces an exact sequence
0→HomG(vPGI, vGPJ)→HomG(IndGPα(vMMαα∩PI), vGPJ)→HomG(vPGIt{α}, vPGJ)
→Ext1G(vPGI, vGPJ)→Ext1G(IndGPα(vMMαα∩PI), vGPJ)→Ext1G(vPGIt{α}, vGPJ).
The adjunction betweenIndGPα andOrdPα yields an isomorphism
HomG(IndGPα(vMMαα∩PI), vPGJ)'HomMα(vMMαα∩PI,OrdPα(vGPJ)).
Moreover, there is a short exact sequence
0→Ext1Mα(vMMαα∩PI,OrdPα(vGPJ))→Ext1G(IndGPα(vMMαα∩PI), vPGJ)
→HomMα(vMMαα∩PI, H1OrdPα(vGPJ))
(see [20, (3.7.6)] if char(F) = 0 and [31, Cor. 2] if char(F) = p). By [2, Th. 6.1(ii)], there is an Mα-equivariant isomorphism
OrdPα(vGP
J)'
(vMMαα∩PJ if α /∈J,
0 otherwise.
Using Lemma 2.9 below and taking into account Proposition 2.5, we obtain the following results.
Caseα ∈J: If F =Qp and dimUα= 1 assume thatε¯◦αe6= 1. There is an isomorphism (2.7) Ext1G(vPGI, vGPJ)'
(R if J =It {α}, 0 otherwise.
In the caseJ =It {α}, theR-module Ext1G(vGP
I, vPG
J) is generated by the class of (2.6).
Caseα /∈J: There is an exact sequence
(2.8) 0→Ext1G(vPGI, vPGJ)→Ext1Mα(vMMαα∩PI, vMMαα∩PJ)→Ext1G(vGPIt{α}, vPGJ).
Lemma 2.9. (1) If H1OrdPα(vGP
J)6= 0 then F =Qp, dimUα = 1, andα∈J. (2) Assume thatF =Qp, dimUα = 1, andα ∈J. Ifε¯◦αe6= 1, then
HomMα(vMMαα∩PI, H1OrdPα(vPGJ)) = 0.
